Methods in Oceanography 8 (2013) 56–74
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Methods in Oceanography
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Full length article
Model of the attenuation coefficient of daily
photosynthetically available radiation in the
upper ocean
Jianwei Wei ∗ , ZhongPing Lee
School for the Environment, University of Massachusetts Boston, 100 Morrissey Blvd., Boston,
MA 02125, USA
article
info
Article history:
Available online 17 December 2013
Keywords:
Photosynthetically available radiation
Diffuse attenuation coefficient
Diurnal variability
Inherent optical properties
∗
abstract
Penetration of the photosynthetically available radiation (PAR, over
400–700 nm) in the upper ocean is important for many processes
such as water radiant heating and primary productivity. Because
of this importance, daily PAR at sea surface (PAR(0+ )) is routinely
generated from ocean-color images for global studies. To propagate this broadband solar radiation through the upper ocean, an
attenuation coefficient of PAR (KPAR ) is also generated from the
same ocean-color measurements. However, due to the empirical
nature of the KPAR algorithm, this KPAR product corresponds to an
instantaneous PAR at a fixed sun angle, with no diurnal variability.
It is hence necessary to have an attenuation coefficient matching
the temporal characteristics of daily PAR. This paper represents an
effort to meet this need. Using ECOLIGHT, the subsurface light field
for a wide range of water bodies was simulated, from which the
attenuation coefficient (KPAR ) of daily PAR was calculated. We presented the diurnal and vertical variation of this attenuation coefficient, and found that it can be well predicted (within ∼7%) as
a function of the total absorption coefficient and backscattering
coefficient at 490 nm and the noontime solar zenith angle. This
new model offers an efficient and reasonably accurate approach for
quantifying daily upper water column PAR within the global ocean
from satellite measurements of water color.
© 2013 Elsevier B.V. All rights reserved.
Corresponding author. Tel.: +1 617 287 7396; fax: +1 617 287 7474.
E-mail addresses: [email protected] (J. Wei), [email protected] (Z.P. Lee).
2211-1220/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.mio.2013.12.001
J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
57
1. Introduction
The visible light (400–700 nm) accounts for about half of the total radiant energy reaching the
ocean surface. Penetration of this broadband solar radiation through the upper ocean is an essential
driver determining many biological, chemical and physical processes (Dickey and Falkowski, 2002).
For instance, thermal and dynamic evolution of the upper ocean is sensitive to the vertical distribution
of available solar energy (Chang and Dickey, 2004; Dickey and Simpson, 1983; Lewis et al., 1990;
Morel and Antoine, 1994; Murtugudde et al., 2002; Ohlmann and Siegel, 2000; Ohlmann et al., 2000;
Siegel and Dickey, 1987a). The water radiant heating rate (RHR) can be defined by the one-dimensional
heating equation (Ohlmann and Siegel, 2000)
dT
dt
=−
1
dPAR
ρw Cp dz
.
(1)
PAR represents the photosynthetically available radiation (in units of W m−2 ) that is an integration
of the irradiance over the spectral range 400–700 nm (a list of symbols and notations are given in
Table 1), ρw is the sea water density, Cp is the specific heat capacity of sea water (∼4100 J kg−1 °C−1 ), T
is the water temperature (unit: °C), and t is time (unit: s). When all the other properties are considered
constant, the heating rate in Eq. (1) is determined by the vertical change of PAR with water depth.
Solar radiation is also the driving force of marine primary productivity. Its value at a depth can be
described by a hyperbolic tangent model (Jassby and Platt, 1976; Platt and Gallegos, 1980)
P (z , t ) = Pmax (z , t ) · tanh

PAR (z , t )
Ek (z , t )

(2)
with PAR(z , t ) here computed in quanta (units: mol photons m−2 s−1 ), Pmax (unit: mg C m−3 h−1 )
the maximum photosynthetic rate, and Ek (unit: mol photons m−2 s−1 ) the light saturation index.
Accurate parameterization of the change of PAR along with water depth is critical for resolving the
depth distribution of primary productivity in euphotic zone (Asanuma et al., 2003; Lohrenz et al.,
1994; Marra et al., 2003; Morel and Berthon, 1989; Smith et al., 1987).
PAR (represented here by downwelling plane irradiance, Ed , in energy units of W m−2 ) at depth z
can be expressed as
PAR (z ) =

700
Ed (z , λ) dλ
(3)
400
and the vertical propagation of Ed is
Ed (z , λ) = Ed (0− , λ) e−Kd (λ) z
(4)
with Kd (λ) the diffuse attenuation coefficient of Ed at wavelength λ. Based on the radiative transfer
theory, it has been found that Kd is a function of water’s inherent optical properties (IOPs) and the solar
zenith angle (Gordon, 1989; Kirk, 1984; Lee et al., 2005a; Loisel and Stramski, 2000; Morel and Gentili,
2004). For vertically homogeneous waters, Kd varies with depth, but generally within ∼10% for low
solar zenith angle and low scattering waters. Following the scheme of Eq. (4), the vertical propagation
of PAR is also commonly expressed as
PAR (z ) = PAR (0− ) e−KPAR z
(5)
with KPAR the vertical attenuation coefficient for PAR. And like Kd , KPAR varies with solar zenith angle
(Lee et al., 2005b). However, unlike the mild vertical variation of Kd , due to the strong selective
absorption by water constituents, KPAR varies strongly with depth and can change by a factor of 3–4
between the surface and deeper depths (Lee, 2009; Paulson and Simpson, 1977; Siegel and Dickey,
1987a).
Many ocean circulation models use daily solar radiation as inputs (Brodeau et al., 2010; Large et al.,
1997; Murtugudde et al., 2002), and the daily total solar radiation is
PAR (z ) =

sunset
sunrise

700
400
Ed (z , λ) dλ dt .
(6)
58
J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
Table 1
Notations and definitions.
Notation
Definition
Units
a
b
bb
CHL
Cp
Ed
Ed (0− )
Ek
Enet
Eo
Eod
Eu
Kd
KPAR
KPAR
Absorption coefficient
Scattering coefficient
Backscattering coefficient
Chlorophyll a concentration
Specific heat capacity of water
Downwelling plane irradiance
Ed right below sea surface
Light saturation index
Downwelling net irradiance
Total scalar irradiance
Downwelling scalar irradiance
Upwelling plane irradiance
Diffuse attenuation coefficient for Ed
Diffuse attenuation coefficient of daily PAR
Diffuse attenuation coefficient of instantaneous PAR
Wavelength
Photosynthetically available radiation
Daily PAR
Daily PAR at right below surface
Maximum photosynthetic rate
Noontime solar zenith angle
Instantaneous solar zenith angle
Radiant heating rate
Solar transmission function
Water depth (positive downward)
Euphotic depth
m−1
m−1
m−1
mg m−3
J kg−1 °C−1
W m−2 , or mol photons m−2
W m−2 , or mol photons m−2
mol photons m−2 s−1
W m−2 , or mol photons m−2
W m−2 , or mol photons m−2
W m−2 , or mol photons m−2
W m−2 , or mol photons m−2
m−1
m−1
m−1
nm
W m−2 , or mol photons m−2
W m−2 , or mol photons m−2
W m−2 , or mol photons m−2
mgC m−3 h−1
λ
PAR
PAR
PAR (0− )
Pmax
θn
θs
RHR
Tr
z
zeu
s−1
s−1
s−1
s−1
s−1
s−1
s−1
s−1
s−1
°
°
°C day−1
Dimensionless
m
m
An expression similar to Eq. (5) can be devised to characterize the propagation of this daily PAR in the
upper water column
PAR (z ) = PAR (0− ) e−KPAR ·z .
(7)
Clearly, it is requisite to have both PAR (0 ) (a daily value just below the water surface) and KPAR
for an accurate description of the daily vertical PAR (z ). The daily PAR just above the ocean surface,
PAR (0+ ), is a standard product for all ocean-color satellite missions (Frouin and Murakami, 2007), and
can be further converted to PAR (0− ) after considering the air–water transmittance (Mobley and Boss,
2012). However, there exist no models or products specifically representing the depth attenuation of
the daily PAR.
Based on numerous field measurements and modeling efforts, an empirical algorithm has been
developed that describes the instantaneous KPAR as a function of chlorophyll concentration (CHL)
(Morel, 1991; Morel and Antoine, 1994; Ohlmann and Siegel, 2000). The empirically derived KPAR ,
however, because of the following reasons, is not applicable to Eq. (7):
−
(1) it is a relationship for KPAR calculated between the water surface and 1% of surface PAR. Because
KPAR varies significantly with depth, adopting a depth-insensitive value will result in significant
errors in the estimated PAR at depth;
(2) the empirical relationship is based on normalized water-leaving radiance, therefore, is insensitive
to diurnal changes in the subsurface light field.
In short, there is no model or product of KPAR that can be used to adequately estimate the daily PAR at
depth with Eq. (7).
This study tries to fill this gap by first characterizing the variation of KPAR of different waters and
then by developing an IOP-based KPAR model. As now products of IOPs of the global oceans can be
routinely generated from advanced ocean-color inverse models (IOCCG, 2006), such a KPAR model
will facilitate the estimation of daily PAR at depth, thus improving studies of water heating budget
as well as primary productivity in the upper water column. The paper is organized such that the
J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
59
characteristics of KPAR will be presented first following a series of numerical simulations, and then
a semi-empirical model of KPAR with IOPs and noontime solar zenith angle as inputs. Finally, the
potential improvements on the heating budget and primary production are discussed. Note that the
study here focuses on days with stable sky conditions (either completely cloud free, or the clouds are
fixed in positions), where the complex impact of varying clouds on the daily PAR is omitted.
2. Data and methods
Since we are interested in the irradiance field and its diffuse attenuation coefficient, the newly
developed ECOLIGHT subroutine (Mobley, 2011) is adopted in the model runs. By resolving the
azimuthally averaged radiative transfer equation, ECOLIGHT takes much less computer time than
solving azimuthally dependent radiative transfer equation. In our simulations, the spectral range
between 400 and 700 nm is considered and distributed at band spacing of 10 nm. Inelastic scattering
(Raman scattering and fluorescence scattering) is excluded because it is negligible to heat transfer and
photosynthesis in the water column (Morel and Gentili, 2004).
Downwelling plane irradiances from both the Sun and sky are computed from the extended
RADTRAN model, which was originally developed by Gregg and Carder (1990). The air pressure at
ground level is 101.24 kPa (equal to 29.9 inHg); aerosol is set as maritime type, with optical thickness
of 0.261 at 550 nm; relative humidity is 80%, water vapor is 2.5 cm; 24-hr wind speed and the current
wind speed are set to 5 m s−1 ; visibility is 15.0 km; total ozone is 300 Dobson units. The sky radiance
distribution is represented by the model of Harrison and Coombes (1988).
The spectral inherent optical properties used in ECOLIGHT runs are adopted from a comprehensive
data set generated by International Ocean Color Coordinating Group (IOCCG, 2006). The IOCCG data
set provides by far the most representative absorption and backscattering coefficients, varying over a
large dynamic range. For example, the absorption coefficient at 490 nm, a(490), varies from 0.020 m−1
to 1.938 m−1 , while the backscattering coefficient bb (490) changes from 0.002 m−1 to 0.131 m−1 ,
covering a variety of oceanic waters. For particle scattering, an averaged Petzold’s phase function is
used (Mobley, 1994), which has a backscattering ratio of 1.8%. The water body is assumed optically
deep and homogeneous with regard to the inherent optical properties, similar to earlier optical
modeling studies (Lee et al., 2005a,b; Morel and Antoine, 1994; Ohlmann and Siegel, 2000).
For the determination of an adequate water layer representing the upper water column, the
euphotic depth (zeu , unit: m), a depth where the instantaneous PAR is 1% of its value at surface (Kirk,
1994), is pre-determined based on the knowledge of inherent optical properties and solar zenith
angles (Lee et al., 2007) before each model run. Although the euphotic depth can get slightly shallower
with the increase of the solar zenith angles (also see Morel and Gentili, 2004), the euphotic depth in
this study is computed for a solar zenith angle θs = 0°.
With the derived euphotic depth, a total of 10 consecutive depths are further defined between
0 and zeu , which are placed at 0.1%, 1%, 2.5%, 5%, 10%, 20%, 35%, 50%, 70%, and 100% of the euphotic
depth, respectively. These water depths are arranged in such a way that the highly variable diffuse
attenuation of PAR at the upper part of the euphotic zone, as revealed in previous studies (e.g. Lee
et al., 2005b; Morel and Antoine, 1994), can be characterized as sufficiently as possible.
The instantaneous PAR changes during the day and peaks at local noon and the pattern of its
temporal evolution in the morning mirrors that of the afternoon (see Appendix). Theoretically, the
daily total PAR can be obtained by integrating the instantaneous PAR from dawn to noon, and doubling
the result
PAR (z ) = 2

noon
PAR (z , t )dt .
(8)
dawn
This integration depends on the photoperiod or length of day, which further varies with the geographic
location (specifically latitude) and the day of the year. To simplify the model simulation and
parameterization, the instantaneous PAR will be integrated with the solar zenith angle instead
PAR (z , θn ) = 2

θn
θ1 =90°
PAR (z , θ )dθ
(9)
60
J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
In Eq. (9), daily PAR is characterized as a function of water depth and the midday solar zenith angle
θn (at 12:00 pm local time) with θ1 = 90° the solar zenith angle at dawn. It is straightforward to
calculate the midday solar zenith angle using simple algorithm, for example, of the method of Kirk
(1994). The radiative transfer simulations are easier to implement with respect to the solar zenith
angle, and convenient to compare with other existing studies (e.g. Lee et al., 2005b; Mobley and Boss,
2012; Morel et al., 2007). Most importantly, although the pattern of the temporal evolution of PAR
with time is slightly different from that of PAR described as a function of the solar zenith angle because
the solar zenith does not change at a constant rate with time during the day, the effect of different
integration schemes of Eqs. (8) and (9) on the diffuse attenuation coefficient of daily PAR is negligible
(see Appendix for further explanation).
The diffuse attenuation coefficient for the daily PAR is estimated as follows
KPAR (z , θn ) = −
1
z

PAR (z )
ln

PAR (0− )

θn


PAR (z , θ )1θ 

1  90°

= − ln  θ

n


z
PAR (0− , θ )1θ
(10)
90°
where PAR (z ) and PAR (0 ) are the daily photosynthetically available radiation derived at water
depth z and the surface, respectively. And the diffuse attenuation coefficient for the instantaneous
PAR is computed with respect to its surface value
−
KPAR (z , θs ) = −
1
z

ln
PAR (z , θs )

PAR (0− , θs )
(11)
where PAR (z , θs ) and PAR (0− , θs ) refer to the instantaneous photosynthetically available radiation
at depth z and the surface, respectively, at time t (corresponding to the solar zenith angle θs ).
An increment of 1θ = 5° in solar zenith angle has been kept for the rest of model runs. There are
9000 model runs carried out in the following study, covering a total of 500 water bodies with a wide
range of water absorption and scattering coefficients and at 18 solar zenith angles. The solar zenith
angles beyond 85° are not considered.
3. Results
3.1. Characteristics of the diffuse attenuation coefficient KPAR
Three ensembles of diffuse attenuation coefficients for the PAR are first exemplified in Fig. 1, where
each group represents a unique water body characterized by its inherent optical properties. In a way
similar to the diffuse attenuation coefficient of instantaneous PAR, the diffuse attenuation coefficient
of the daily PAR decreases sharply with depth near the surface and then gradually transitions to a
slower rate of change with depth. This surface-depth transition can be very large; for example, KPAR
could be reduced by a factor of ∼3 within the first 10 m of the water column (See group (a) in Fig. 1).
Magnitude of KPAR is largely determined by the water IOP’s: larger absorption and backscattering
coefficients will result in higher diffuse attenuation for the daily PAR.
The diffuse attenuation coefficient of instantaneous PAR, KPAR , can vary by over 20% when solar
zenith angle θs increases from 0° to 60°. The diffuse attenuation coefficient of daily PAR, KPAR , falls
somewhere between that of the instantaneous PAR at θs 0° and θs = 60°. The difference between KPAR
and KPAR can still be significant as suggested by Fig. 1, depending on the solar zenith angle selected.
The ranges of their percent difference, of the same three waters as shown in Fig. 1, is summarized in
Table 2, where the solar zenith angle θs ≤ θn ≤ 60° is considered. The maximum difference (>16%)
occurs in surface waters, while the minimum difference is often found at the base of the euphotic
zone.
J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
61
Fig. 1. Diffuse attenuation coefficient for daily PAR and instantaneous PAR (in energy units) in euphotic layers. PAR and daily
PAR are integrated from downwelling plane irradiance according to Eqs. (3) and (6), respectively.
Table 2
Relative percent difference between KPAR and KPAR * .
Solar zenith angle at noon θn
Depth (relative to zeu )
Type (a)tblfn**
a = 0.022 m−1
bb = 0.003 m−1
Type (b)tblfn**
a = 0.090 m−1
bb = 0.007 m−1
Type (c)**
a = 0.201 m−1
bb = 0.019 m−1
0°
1%
10%
50%
100%
−9.1%–15.3%
−6%–10.5%
−5.7%–10.1%
−4.4%–7.6%
−9.6%–15.8%
−8.4%–14.1%
−6.3%–11%
−4.9%–8.4%
−10.1%–16.4%
−9.6%–16.1%
−6.2%–10.4%
−3.9%–6.3%
30°
1%
10%
50%
100%
−9.7%–7.3%
−6.4%–5.2%
−5.8%–4.9%
−4.3%–3.6%
−10.3%–7.3%
−8.9%–6.6%
−6.5%–5.2%
−4.8%–3.9%
−10.7%–7.5%
−10.1%–7.4%
−6.2%–4.8%
−3.8%–2.9%
**
*
Three water types (a), (b) and (c) are the same with Fig. 1.
The daily PAR is integrated from downwelling plane irradiance Ed (in units of W m−2 ).
The integration of the diurnally variable PAR removes the dependence on solar zenith angle θs . But
the resulting KPAR for the daily PAR is apparently further determined by the midday solar zenith angle,
θn , which changes with both the latitude and date of year (Kirk, 1994). For example, from θn = 0° to
θn = 60°, KPAR at surface increases by 17% (water type (a) in Fig. 1), 21% (water type (b) in Fig. 1), and
62
J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
Fig. 2. Derived KPAR from KPAR (θn ) following Eq. (12) versus ECOLIGHT generated KPAR . PAR is integrated from downwelling
plane irradiance according to Eq. (6).
22% (water type (c) in Fig. 1), respectively. This result is consistent with the radiative transfer theory:
lower sun increases the optical path length such that the energy of spectral irradiance diminishes
faster with respect to water depth.
Apart from the differences between the two types of attenuation coefficients mentioned above,
the KPAR is also related to KPAR , which can be described in a relatively simple analytical form. For
example, an empirical relationship between the attenuation coefficient of the daily PAR and that of
instantaneous PAR of all depths is derived using least-square regression, as below,
KPAR = 1.060 · [KPAR (θn )]1.003
θn ≤ 70°
(12)
where KPAR (θn ) specifically refers to the attenuation coefficient for instantaneous PAR at noon
(12:00 pm local time). Eq. (12) is based on data with noontime solar zenith angle less than 70°.
According to this equation, an excellent prediction of the KPAR is achievable from the knowledge of
KPAR (θn ) (R2 = 0.99, with mean percentage error 3.6% and root mean square error 2.6%); such a data
comparison is given in Fig. 2, where the y-axis is modeled from Eq. (12) and x-axis represents the
ECOLIGHT-generated KPAR . The mean error between the modeled and ECOLIGHT generated data is
only about 2.8%, with a maximum error of 8%.
3.2. Model KPAR as a function of a, bb and θn
The model given in Eq. (12) is further expressed by a two-parameter model using nonlinear leastsquare regression, which gives the best characterization of the depth-averaged diffuse attenuation
coefficient for the daily PAR in the euphotic zone. Specifically, the part within the square bracket takes
a form similar to the model of Lee et al. (2005b), which was originally developed for the instantaneous
PAR quantified for the downwelling plane irradiance,

KPAR (z ) = 1.060 · K1 +
K2
(z + 0.5)0.45
1.003
.
(13)
The constant 1.060 and the power 1.003 are the same with that of Eq. (12). The model parameters
K1 and K2 , as well as the constants appearing in the model, are derived by least-square fitting to
the ECOLIGHT generated KPAR depth profiles. For example, model parameters K1 and K2 increase
monotonically with the absorption coefficient a(490) and backscattering coefficient bb (490), and
can be described by an appropriate power function, respectively (Fig. 3). According to the data,
J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
63
Fig. 3. Model parameters K1 and K2 varying with absorption coefficient and backscattering coefficient.
the absorption coefficient and backscattering coefficient at wavelength 490 nm provide the best
description for K1 and K2 (with the least percentage error) among the visible bands, which are further
modeled as (also see Lee et al., 2005b)
K1 = A1 + A2 · [a(490)]0.5 + A3 · bb (490)
(14)
K2 = B1 + B2 · a(490) + B3 · bb (490)
(15)
where the two sets of model fitting parameters A and B are depth-independent, but vary with the
noontime solar zenith angle, and it was found that a linear function of cos(θn ) is adequate for
characterizing the variability of parameters A and B. Table 3 presents the resulted coefficients after
least-square fitting.
Fig. 4 compares the ECOLIGHT generated KPAR data with the modeled KPAR based on Eqs. (13)–(15)
and Table 3. A very
agreement is achieved
with a mean absolute percentage error of 7.0%, derived

N  good
model
data
data
as, ε = N1
|
K
−
K
|/
K
×
100%.
The errors for the modeled daily PAR at the bottom
n
n
n
n=1
of euphotic layer are a bit larger, but still less than 9.0% with respect to the ECOLIGHT data.
3.3. Parameterization of KPAR quantified by various irradiance
We have hitherto examined the daily PAR integrated from the downwelling plane irradiance
(in energetic units) according to Eq. (6). In practice, the photosynthetically available radiation is
sometimes represented by the downwelling net irradiance (Enet ), downwelling scalar irradiance (Eod )
or total scalar irradiance (Eo ), depending on the preference of the practitioners and the aimed scientific
questions (Large et al., 1997; Mobley and Boss, 2012; Morel and Antoine, 1994; Morel and Gentili,
2004). For example, the downwelling net irradiance in units of W m−2 is preferably used in Eq. (1);
64
J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
Table 3
Models for the depth-averaged diffuse attenuation coefficient of the daily PAR and the corresponding model parameters.
Modela
Parameter
(1) Downwelling plane irradiance and net irradiance in energy units

KPAR (z ) = 1.060 · K1 +
1.003
K2
(z +0.5)0.45
(2) Downwelling scalar irradiance and total scalar irradiance in energy

units KPAR (z ) = 1.088 · K1 +
K2
1.005
(z +0.5)0.2
(3) Downwelling plane irradiance and net irradiance in quantum units

KPAR (z ) = 1.060 · K1 +
K2
1.003
(z +0.5)0.45
(4) Downwelling scalar irradiance and total scalar irradiance in

quantum units KPAR (z ) = 1.092 · K1 +
a
K2
(z +0.5)0.1
1.007
A1
A2
A3
B1
B2
B3
A1
A2
A3
B1
B2
B3
Coefficients of linear regression
Slope
Intercept
−0.012
−0.061
−0.065
1.365
−0.040
−0.203
−2.030
0.691
3.092
0.135
0.580
−0.791
0.000
−0.101
−0.065
0.405
8.762
0.193
0.816
−8.106
2.088
−0.039
−0.224
−3.070
A1
A2
A3
B1
B2
B3
−0.003
−0.085
A1
A2
A3
B1
B2
B3
0.030
0.000
5.522
−0.087
−0.293
−6.321
1.376
−0.052
−0.161
−2.048
−0.082
0.746
3.042
0.176
0.421
−0.755
−0.274
0.184
14.114
0.437
0.981
−13.973
Note that the model constants may vary for different types of irradiance.
Fig. 4. Comparison of modeled KPAR with ECOLIGHT data according to Eqs. (13)–(15) and Table 3. The daily PAR is integrated
from downwelling plane irradiance (in energetic units) according to Eq. (6).
on the other hand, the total scalar irradiance in unit of mol photons m−2 s−1 is more suitable for
the estimation of primary production in Eq. (2). A question may arise immediately: whether or not
the daily PAR quantified by various forms of irradiance will significantly impact the attenuation
coefficient.
J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
65
Table 4
Matrix of the symmetric mean absolute percentage error (SMAPE) for the diffuse attenuation coefficient of daily PAR quantified
by various types of irradiance.
Type of irradiance
(1)
(2)
(3)
(4)
(5)
(6)
(7)
1. Ed in energy units
0
2. Enet in energy units
1.8%**
3. Eo in energy units
13.4%
15.2%
0
4. Eod in energy units
10.7%
12.5%
2.8%
0
5. Ed in quantum units*
4.3%
4.0%
15.9%
13.2%
0
6. Enet in quantum units
5.1%
4.3%
17.7%
15.0%
1.8%
0
7. Eo in quantum units
11.3%
12.8%
4.8%
4.3%
13.2%
15.0%
0
8. Eod in quantum units
9.6%
10.7%
6.1%
4.8%
10.5%
12.3%
2.8%
*
(8)
0
The daily PAR in quantum units is integrated from irradiance according to the following, PAR (z , θn )
0
=
 noon  700
E (z , λ)λh−1 c0−1 Na−1 dλ dt, where h is the Planck constant, c0 is the light speed, Na is the Avogadro number, and E refers to the irradiance Ed , Enet , Eo or Eod in energy units.
**
Shading is to highlight the small errors (<5%), specifically between different types of irradiance.
1
12
sunrise
400
To find an answer, the symmetric mean absolute percentage error (SMAPE) is quantified among
KPAR calculated from all types of irradiance obtained from ECOLIGHT simulations, which is derived as
ε =
2
N
N

n =1


|Kn1 − Kn2 |/ Kn1 + Kn2 × 100%. It is found that the diffuse attenuation coefficient of
the daily PAR derived from the downwelling plane irradiance is almost equivalent to that from the
downwelling net irradiance, with a mean difference of less than 5% (see Table 4). Such small errors
are also found in the KPAR derived from the downwelling scalar irradiance and total scalar irradiance.
This reflects the fact that the downwelling light is typically orders of magnitude brighter than the
upwelling light in ocean waters. Between the downwelling plane irradiance and downwelling/total
scalar irradiance, however, larger differences (>10%) are found frequently between the KPAR
calculated from downwelling irradiance and the KPAR calculated from downwelling/total scalar
irradiance. Another interesting finding is on the usage of different units, i.e. energy versus quanta. That
is, the diffuse attenuation coefficient KPAR quantified for an energetic irradiance is almost the same as
that of its quantum counterpart (with a difference less than 5%), which is consistent with the results
of Morel (1988) and Morel and Gentili (2004) when analyzing the diffuse attenuation coefficient of
instantaneous PAR.
It has been found that the attenuation coefficient of daily PAR as estimated from various types
of irradiance can be modeled in a form similar to Eq. (12), with slightly different model constants
(see Table 3). Based on this fact, we have developed KPAR models similar to Eq. (13) (and the same
with Eqs. (14) and (15)) to better account for the large variance among different irradiance forms. The
models together with their model coefficients are tabulated in Table 3. It is noted that model (1) and
model (3) in Table 3 have taken exactly the same formulation, but with different model coefficients.
Fig. 5 shows the model-data comparison with respect to various irradiance forms, where an absolute
percentage difference generally within ∼7% is achieved.
4. Discussion
Light is diminished due to absorption and scattering effects when propagating in the water mass
(Jerlov, 1976). Generally water molecules contribute strong attenuation in the red wavelengths while
the dissolved and suspended constituents contribute varying attenuation in the blue bands. This
wavelength selective attenuation (or selective absorption) leads to the fast extinction of short- and
long-wavelength radiation in very surface layers of the water column. The spectral radiation is thus
narrowed to blue–green wavebands towards depth, and therefore the diffuse attenuation coefficient
for such broad band radiation will vary significantly with depth. Lower solar elevation (larger solar
zenith angle) further increases the optical path lengths of photons in the water, leading to the faster
extinction with respect to the water depth, and hence a higher diffuse attenuation coefficient (Gordon,
1989; Kirk, 1984). KPAR is deliberately modeled as a function of water depth and midday solar zenith
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J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
Fig. 5. Comparison of modeled KPAR with ECOLIGHT data. The type of irradiance quantified for the daily PAR is indicated in
each subplot. The exact formulation for each model is summarized in Table 3.
angle to account for these impacts. Further the absorption coefficient a(490) and backscattering
coefficient bb (490) of the upper water column, instead of chlorophyll concentration, are used as
inputs. The band 490 nm is chosen because the light at this particular band is historically measured and
generally penetrates deeper in oceanic water column than most of the other bands, and because the
inherent optical properties are derivable from ocean-color measurements (Zaneveld et al., 2006) and
with robust accuracy (Lee et al., 2002; Loisel and Stramski, 2000; Maritorena et al., 2002; Smyth et al.,
2006). One critical component of the KPAR model, as suggested by Eq. (12), is a model for the diffuse
attenuation coefficient of the midday instantaneous PAR. The KPAR (θn ) in this study is described by
a function almost the same as in Lee et al. (2005b) in view of the way incorporating the water IOP’s,
but specific to the noontime light field. With this caveat in mind, it is found that a direct substitution
of model of Lee et al. (2005b) into Eq. (12) will also give an accurate estimation of KPAR (figure not
shown), as far as the downwelling plane irradiance and net irradiance are concerned.
4.1. Model error and sensitivity
Like many other studies involving solar radiation, the daily PAR in this study represents a temporal
integration of the instantaneous PAR. It differs from the average daily PAR (which will be a temporal
mean) by a constant scalar. So the average and the total daily PAR, no matter whichever is used
in a specific context, have the same diffuse attenuation coefficient. The solar zenith angle is used
as a surrogate to time in Eq. (8). Although not exactly equal to each other, the adaptation of fine
J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
67
Fig. 6. Sensitivity of model output KPAR to model parameters a(490), bb (490) and θn . The water optical properties same with
water type (a) in Fig. 1, and noontime solar zenith at 30° are chosen as the baseline parameters.
resolution (1θ = 5°) has guaranteed the satisfactory estimation for the daily PAR, which is equivalent
to a temporal resolution better than ∼1 h. A new parameter of the noontime solar zenith angle is
introduced into the KPAR model. This is advantageous over other parameters in that its computation
is very straightforward and fairly accurate with a priori knowledge of the location (latitude) and the
date (day of year) (e.g. Kirk, 1994; Reda and Andreas, 2004). Thus, it is very convenient to apply this
model from location to location, and day to day.
This KPAR model is developed out of IOCCG data set. As a result, it is most applicable to clear oceanic
waters. In optically complex waters, such as the coastal turbid water, the presence of mineral particles
becomes important and may cause different optical characteristics than that of the clear oceanic water
(e.g. Babin and Stramski, 2004).
A sensitivity analysis is performed to assess the effects of error in other optical properties on the
modeled diffuse attenuation coefficient KPAR . A series of percentage changes are implemented with
regard to a(490) and bb (490), respectively. As illustrated in Fig. 6, the model output KPAR is most
sensitive to the absorption coefficient a(490) throughout the euphotic zone. This is evidenced by the
sharpest change of the absolute value of the model output. The role of the backscattering coefficient is
second to the absorption coefficient at the upper part of the euphotic layer. The effect of solar zenith
angle θn becomes less important with the increasing depth. This agrees with the empirical relationship
between a, b, and Kd as reported by Kirk (1984).
We also evaluated the impact of particle phase function on the KPAR model. Additional ECOLIGHT
model runs were implemented with a Fournier–Forand phase function, which is given by an analytical
form and allows the phase function to be altered with different backscattering ratios (Fournier and
Forand, 1994). Keeping the particle scattering coefficient the same, KPAR may be enhanced by as large
as 37% when the backscattering ratio (b̃bp = bb/b) is raised from 0.2% to 5%, simply because of an
increased backscattering coefficient. Nevertheless, good agreements can still be achieved between
the ECOLIGHT KPAR data and the modeled KPAR from Eq. (13), with mean error of less than 12%. For the
commonly found b̃bp in natural waters, roughly between 1% and 2% (Whitmire et al., 2010), the mean
error of KPAR model is always less than 10%.
The above discussion has focused on the ideal scenario of clear skies, ignoring the potential effects
of clouds, which are often present over the oceans. Clouds can greatly alter the spectral radiance
and irradiance incident on the ocean surface (Siegel et al., 1999), and further potentially impact the
attenuation of the PAR within the upper ocean water column. We assessed the degree of the effect
of clouds on the attenuation coefficient KPAR in the euphotic zone based on ECOLIGHT simulations.
We found that KPAR of the daily PAR is generally quite insensitive to the cloudiness (with the current
cloud model) and the cloud-induced change in KPAR is generally less than 5%. Theoretically, KPAR may
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J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
be occasionally subject to a change of 10% when the following conditions are reached simultaneously:
the sky is completely overcast, midday solar zenith is very large (>70°), and the water itself is highly
scattering. This small effect of clouds on the attenuation coefficient of the daily PAR is similar to the
spectral diffuse attenuation coefficient reported before (Kirk, 1984; Siegel and Dickey, 1987b).
The data presented so far are based on a specific sky with τa (550) = 0.261, ozone = 300 DU, WV =
2.5 cm, and aerosol type 1 (maritime aerosol as used in ECOLIGHT). Similar to the effect of clouds,
variations in these atmospheric factors will cause variable sky radiance distribution and underwater
radiance distribution. To test the sensitivity of KPAR to the variation of atmospheric conditions, the
atmospheric parameters are varied one by one in a series of new ECOLIGHT simulations, from 94.98
to 101.99 kPa (28.05 to 30.12 in Hg) for air pressure, 1 to 6 cm for water vapor, 20% to 90% for relative
humidity, 5.5 to 18 km for visibility, 200 to 450 DU for total ozone, and aerosol type 1, 2 and 3. The
analyses indicate that the difference due to the variation of atmospheric parameters is negligibly
small. The biggest difference in the resultant KPAR (for three sea waters of type (a), type (b) and
type (c)) is only about 1.35%.
4.2. Satellite estimation of the solar transmission in global oceans
Remotely sensed ocean color data is widely used to parameterize the biogenic heating in different
levels of ocean circulation models (Ballabrera-Poy et al., 2007; Kara et al., 2005; Murtugudde et al.,
2002; Ohlmann, 2003; Sweeney et al., 2005). For example, the satellite ocean chlorophyll climatology
is often adopted for the derivation of the diffuse attenuation coefficient of the solar radiation based
on the bio-optical models (Morel, 1988; Morel and Antoine, 1994; Morel et al., 2007), and further
the derivation of solar transmission in the water column. The solar transmission is the ratio of solar
radiation at depth z to that just below the water surface or the fraction of surface solar radiation left
at depth z (Lee et al., 2005b; Morel and Antoine, 1994; Ohlmann and Siegel, 2000; Siegel et al., 1995),
and is a critical component in water heating modeling (see Ohlmann, 2003 and references therein). It
is formulated as below
Tr (z ) = exp(−K · z )
(16)
where Tr is the solar transmission, and K is the attenuation coefficient of solar radiation. Since K is
depth dependent in the upper water column, it is important to appropriately represent the penetration
of solar radiation, according to the time scales (for example, daily PAR versus instantaneous PAR)
and/or spatial scales (for example, depth-dependent versus depth-independent) being discussed.
Contrasting examples of solar transmission in ocean waters are illustrated in Fig. 7. The climatology
of solar transmission in the North Atlantic Ocean is characteristic of great spatial variability, no
matter whatever time scales are focused. According to the specific example, however, use of the
diffuse attenuation KPAR will likely underestimate the daily solar transmission. The relative percentage
difference ((TrPAR − TrPAR )/TrPAR × 100%) is found around 50% for most pixels in the central North
Atlantic (Fig. 8(a)). It seems that the relative difference in wintertime has dropped to some extent,
probably because the difference between KPAR and KPAR is smaller when the noontime solar zenith gets
larger (austral summer relative to austral winter). In spite of possible reasons causing this difference,
a 50% of error in the solar transmission could be significant in the modeling of the ocean dynamics
and biogeochemistry.
It is not uncommon in ocean models to consider the depth-dependent irradiance with a fixed
attenuation coefficient (e.g. Kara et al., 2005; Schneider and Zhu, 1998; Wetzel et al., 2006), which
allows the solar irradiance to penetrate deeper within the water. When interpreting the modeling
results, it is always worthwhile to note the potential difference induced by using depth-fixed versus
depth-evolving diffuse attenuation coefficients, as the percentage difference between KPAR at two
consecutive depths could be very large. In Fig. 8(b), for example, the diffuse attenuation coefficient
at the first optical depth (∼1/Kd (490)) can be 70% smaller than that at the second optical depth
(2/Kd (490)).
It is also noted that the diffuse attenuation coefficient for the daily PAR is often larger than that of
the instantaneous PAR around noontime (e.g. Fig. 1). In theory, the daily solar transmission is most
likely less than the instantaneous solar transmission around noontime. The opposing observations
J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
69
Fig. 7. Daily solar transmission in North Atlantic Ocean and Arctic Ocean in summertime (July 12, 2012). The left-column
panels (a and c) are results derived based on the instantaneous attenuation coefficient; the right-column panels (b and d) are
derived from KPAR model. Panels (a) and (b) refer to the water column from surface to depth of 1/Kd (490); panels (c) and (d)
correspond to the water column down to depth 2/Kd (490). Panels (a) and (b) are derived from the Eqs. (9) and (9′ ) of Morel
et al. (2007), with the MODIS monthly products of the diffuse attenuation coefficient for instantaneous PAR at 490 nm as input.
KPAR is computed for daily PAR in energetic units based on the model developed in this study; a(490) and bb (490) are estimated
with QAA v5.0 based on the remote-sensing reflectance.
given in Fig. 7 are possibly a result of misrepresenting the diffuse attenuation coefficient for the
instantaneous PAR, which is currently related to chlorophyll a concentration (Morel et al., 2007).
Although the KPAR model has yet to be validated against in-situ measurements, it represents a
more meaningful formulation of the solar transmission of the photosynthetically available radiation.
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J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
Fig. 8. (a) Percentage difference between the estimated daily solar transmission at water depth 1/Kd (490) (calculate as
ε = |TrPAR − TrPAR /TrPAR | × 100%). (b) Percentage difference of the daily solar transmission at depth z1 = 1/Kd (490) versus
z
z
z
z2 = 2/Kd (490), which is calculated as ε = (|Tr 1 − Tr 2 |)/Tr 1 × 100%.
PAR
PAR
PAR
Another advantage with KPAR model is that its inputs (a(490) and bb (490)) are derivable from the
ocean-color images of the global ocean and with higher accuracy than the indirect estimation of
chlorophyll concentration. It is pertinent to and likely operational for ocean modeling to account for
the depth-evolving diffuse attenuation coefficient for daily PAR in future practice.
4.3. Potential effects of KPAR model on radiant heating and carbon budget
The new KPAR model can be easily applied to studies of the water radiant heating rate and water
column primary production, under the condition that the water absorption coefficient a(490) and
backscattering coefficient bb (490) have been derived from ocean-color images and that they do not
change in a day. Two scenarios related to KPAR are described below. Keep in mind that the situations
depicted here are over-simplistic. Other phenomena occurring in the ocean could complicate these
processes. For example, the presence of turbulence in the upper ocean generates fluctuations in
temperature and salinity (Xu et al., 2012) and influences the growth rate of phytoplankton (Durham
et al., 2013).
The daily water radiant heating rate may be estimated by rewriting Eq. (1) and substituting KPAR
in the following relation
RHR (z ) =


PAR (0− ) 
· 1 − exp −KPAR · z .
ρ w Cp z
(17)
According to the profiles given in Fig. 9(a), an error of ∼15% is expectable in the upper layers of water
column, if using KPAR instead of KPAR . The error is often minimal at the bottom of the euphotic layer.
The relatively larger errors found at shallower depths are likely to have an important effect on the
thermal structure and heat balance in the mixed layers of the ocean where small temperature errors
can be climatically important (Lewis et al., 1990; Murtugudde et al., 2002).
Another application of the KPAR model is for the estimation of the daily water-column primary
production. Fig. 9(b) illustrates the depth profiles of the water column primary production, by
assuming constant saturation index and maximum photosynthetic rate in Eq. (2). According to
this example, using the attenuation coefficient of instantaneous PAR can overestimate the primary
production by 25% compared to its ‘‘true’’ value. If taking into account the fact that the parameters
of Ek and Pmax are also subject to diurnal variability (Smith et al., 1989), the resulted errors could be
J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
71
Fig. 9. (a) Profiles of water radiant heating rate. (b) Profiles of daily primary production. The water body examined here has
the same inherent optical properties with water type (b) in Fig. 1.
larger. Besides the diurnal variability, the saturation index and maximum photosynthetic rate can also
be variable along the water depth. So the estimated production shown in Fig. 9(b) is likely unrealistic
and arguable in itself. How to parameterize a photosynthetic rate, which is adapted to the daily PAR
field, is beyond the scope of the current study.
5. Concluding remarks
In this study, based on the radiative transfer, models for the diffuse attenuation (KPAR ) of daily
photosynthetically available radiation in the euphotic layer are developed for the first time. This
KPAR model can then be applied to ocean-color images to propagate the surface daily PAR product
to deeper depths for the studies of heating rate and primary production; where the traditionally
estimated diffuse attenuation coefficient KPAR (an instantaneous product) is not exactly applicable.
Similar to the KPAR model for instantaneous PAR of Lee et al. (2005b), the model is described as a
function of the absorption coefficient a(490) and backscattering coefficient bb (490) and the solar
zenith angle at noon. The new model provides a robust estimate for the diffuse attenuation coefficient
of the daily photosynthetically radiation in clear oceanic waters. Except for the model residual errors
(7%), the uncertainty of the model prediction in practice will depend on the accuracy of model inputs,
particularly the absorption coefficient and backscattering coefficient.
Our model assumed a homogeneously mixed euphotic zone which contains no significant vertical
structures in the inherent optical properties. If absorption and backscattering coefficients were to vary
significantly along the water depth, the pattern of diffuse attenuation coefficient profiles of PAR could
deviate to some degree from our results. In addition, effects of clouds were not considered, as it could
be highly variable in many places.
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J. Wei, Z. Lee / Methods in Oceanography 8 (2013) 56–74
Fig. A.1. Evolution of the instantaneous photosynthetically available radiation during the day (day of year: 90, latitude: 20°N)
at (a) right below the water surface and (b) the bottom of euphotic layer (∼110 m). Time step used is 1t = 0.25 h; zenith angle
resolution is 1θs = 5°. The water optical properties are a(490) = 0.0223 m−1 and bb (490) = 0.0031 m−1 .
Acknowledgments
This study was funded by National Aeronautic and Space Administration (NASA) Ocean Biology
and Biogeochemistry and Water and Energy Cycle Programs, the JPSS VIIRS Ocean Color Cal/Val Project
and the Royal Belgian Institute of Natural Sciences, Belgium. The authors would like to thank Dr. Kevin
Ruddick, Dr. Steve Ackleson and two anonymous reviewers for their comments and suggestions which
have greatly improved this manuscript.
Appendix
To validate model of Eq. (9), additional Hydrolight model runs were made with regard to selected
locations (0°N, 20°N, and 40°N in latitude), day of year (1, 90 and 180), and water bodies of different
optical properties (water type (a): a(490) = 0.022 m−1 and bb (490) = 0.003 m−1 , water type (b):
a(490) = 0.090 m−1 and bb (490) = 0.007 m−1 , and water type (c): a(490) = 0.201 m−1 and
bb (490) = 0.019 m−1 , respectively). Fig. A.1 illustrates the daily evolution of the instantaneous
PAR with resolution of 1t = 0.25 h in time and 1θs = 5° in terms of zenith angle, respectively.
Because the relationship between the solar zenith and time of day is not exactly linear: it evolves
faster near dawn/dusk and slower around noon, PAR changes at a faster rate with solar zenith angle
θs , particularly when θs is small (near noontime). In consequence, the PAR–angle integration scheme
and PAR–time integration scheme will not give the same estimation for the daily PAR. However, it is
the ratio of the daily PAR at two water depths, not their respective absolute value that determines the
diffuse attenuation coefficient KPAR (recall Eq. (7)). It was found that the ratio of daily PAR at two water
depths remain approximately the same for both integration schemes. And the resultant KPAR within
the euphotic layer is insensitive to the specific integration scheme (time versus angle), with a relative
error of less than 2%. It is also worth emphasizing that KPAR is not sensitive to the solar zenith angle
resolution 1θ either, particularly when the solar zenith is varied at a small step. According to the test
runs, resolutions of 1°, 2.5°, 5°, and 7.5° all give about the same estimation for KPAR (1% difference).
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